Spectral Graph Analysis: A Unified Explanation and Modern Perspectives

TL;DR

This paper seeks to unify spectral graph analysis under a single statistical framework, enhancing understanding and practical algorithm design for complex networks across various fields.

Contribution

It introduces a universal statistical logic that explains and encompasses nearly all spectral graph methods within one formalism.

Findings

Proposes a simple, universal spectral graph analysis framework

Reveals underlying statistical principles of spectral methods

Facilitates improved algorithm development for network analysis

Abstract

Complex networks or graphs are ubiquitous in sciences and engineering: biological networks, brain networks, transportation networks, social networks, and the World Wide Web, to name a few. Spectral graph theory provides a set of useful techniques and models for understanding `patterns of interconnectedness' in a graph. Our prime focus in this paper is on the following question: Is there a unified explanation and description of the fundamental spectral graph methods? There are at least two reasons to be interested in this question. Firstly, to gain a much deeper and refined understanding of the basic foundational principles, and secondly, to derive rich consequences with practical significance for algorithm design. However, despite half a century of research, this question remains one of the most formidable open issues, if not the core problem in modern network science. The achievement of this paper is to take a step towards answering this question by discovering a simple, yet universal statistical logic of spectral graph analysis. The prescribed viewpoint appears to be good enough to accommodate almost all existing spectral graph techniques as a consequence of just one single formalism and algorithm.